According to legend, Pythagoras discovered the foundations of musical tuning by listening to the sounds of four blacksmith's hammers, which produced consonance and dissonance when they were struck simultaneously. According to Nicomachus in his 2nd century CE Enchiridion harmonices  Pythagoras noticed that hammer A produced consonance with hammer B when they were struck together, and hammer C produced consonance with hammer A, but hammers B and C produced dissonance with each other. Hammer D produced such perfect consonance with hammer A that they seemed to be "singing" the same note. Pythagoras rushed into the blacksmith shop to discover why, and found that the explanation was in the weight ratios. The hammers weighed 12, 9, 8, and 6 pounds respectively. Hammers A and D were in a ratio of 2:1, which is the ratio of the octave. Hammers B and C weighed 8 and 9 pounds. Their ratios with hammer D were (12:8 = 3:2 = perfect fifth) and (12:9 = 4:3 = perfect fourth). The space between B and C is a ratio of 9:8, which is equal to the musical whole tone, or whole step interval (Play 9/8).
|A||6||Play 1/1||Play P1||C|
|B||8||Play 3/2||Play P5||G|
|C||9||Play 4/3||Play P4||F|
|D||12||Play 2/1||Play P8||C'|
The legend is, at least with respect to the hammers, demonstrably false. It is probably a Middle Eastern folk tale. These proportions are indeed relevant to string length (e.g. that of a monochord) — using these founding intervals, it is possible to construct the chromatic scale and the basic seven-tone diatonic scale used in modern music, and Pythagoras might well have been influential in the discovery of these proportions (hence, sometimes referred to as Pythagorean tuning) — but the proportions do not have the same relationship to hammer weight and the tones produced by them. However, hammer-driven chisels with equal cross-section, show an exact proportion between length or weight and Eigenfrequency.
Earlier sources mention Pythagoras' interest in harmony and ratio. Xenocrates (4th century BCE), while not as far as we know mentioning the blacksmith story, described Pythagoras' interest in general terms: "Pythagoras discovered also that the intervals in music do not come into being apart from number; for they are an interrelation of quantity with quantity. So he set out to investigate under what conditions concordant intervals come about, and discordant ones, and everything well-attuned and ill-tuned." Whatever the details of the discovery of the relationship between music and ratio, it is regarded as historically the first empirically secure mathematical description of a physical fact. As such, it is symbolic of, and perhaps leads to, the Pythagorean conception of mathematics as nature's modus operandi. As Aristotle was later to write, "the Pythagoreans construct the whole universe out of numbers".
- Weiss, Piero, and Richard Taruskin, eds. Music in the Western World: A History in Documents. 2nd ed. N.p.: Thomson Schirmer, 1984. 3. ISBN 9780534585990.
- Kenneth Sylvan Guthrie, David R. Fideler (1987). The Pythagorean Sourcebook and Library: An Anthology of Ancient Writings which Relate to Pythagoras and Pythagorean Philosophy, p.24. Red Wheel/Weiser. ISBN 9780933999510.
- Christensen, Thomas, ed. The Cambridge history of Western music theory. Cambridge: Cambridge University Press, 2002. 143. ISBN 9780521623711.
- Burkert, Walter (1972). Lore and Science in Ancient Pythagoreanism, p.375. ISBN 9780674539181. Cited in Christensen 2002, p.143.
- Markus Bautsch: Über die pythagoreischen Wurzeln der gregorianischen Modi, English: About the Pythagorean roots of the Gregorian modes, Mater Dolorosa (Berlin-Lankwitz), March 2012, retrieved on 31 August 2017
- Barker (2004). Andrew (ed.). Greek musical writings (1st pbk. ed.). Cambridge: Cambridge University Press. p. 30. ISBN 978-0-521-61697-3.
- Lucas N.H. Bunt; Phillip S. Jones; Jack D. Bedient (1988). The historical roots of elementary mathematics (Reprint ed.). New York: Dover Publications. p. 72. ISBN 978-0-486-25563-7.
- Christian, James (26 January 2011). Philosophy An Introduction to the Art of Wondering. Wadsworth Pub Co. p. 517. ISBN 978-1-111-29808-1.
- Waterfield, transl. with commentary by Robin (2000). The first philosophers : the Presocratics and Sophists (1. publ. as an Oxford world's classics paperback ed.). Oxford: Oxford Univ. Press. p. 103. ISBN 978-0-19-282454-7.
the pythagoreans construct the whole universe out of numbers.